122 VI HUBERT SPACES

4. Let X be a set, E a vector subspace of Cx, on which is given a structure of complex
Hilbert space. A mapping (x, y) -> K(;c, y) of X x X into C is called a reproducing
kernel for E if it satisfies the two following conditions: (1) for every y e X, the function
K(. , y) : x -» K(x, y) belongs to E; (2) for any function fe E, and any y e X, f(y) ==

(a) Show that K is a mapping of positive type of X x X into C, i.e. , for any integer
n ^ 1 and any finite sequence (xOi^^n of points of X, the mapping

of C2" into C is a positive hermitian form. This in particular implies K(x, x) ^ 0 for
every xeX, K(y, x) = K<*, jO and |K(x, ^)|2 ^ K(x, x)K(.y, j>) for x, >> in X. Show
that for /e E, one has |/Cx)| ** \\f\\ ' (K(y, y))112 for y e X.

(b) Show that if (fn) is a sequence of functions of E which converges (for the Hilbert
space structure) to /e E, then, for every x e X, the sequence (fn(xj) converges to f(x)
in C ; the convergence is uniform in any subset of X where the function jc -> K(x, x) is
bounded.

(c) Let (xi)i^i^n be a finite sequence of points of X, (a^i *=* <n a sequence of n com-
plex numbers. Suppose det(K(*{ , */)) 7^ 0, so that the system of linear equations

CjK(xt , jcj) = at (1 < i^ n) has a unique solution (cj). Show that among the func-
i «

tions /e E such that /(;tj) = a{ for 1 < z < /z, the function /o = T! cjK(., .*/) has the

j = i

smallest norm. In particular, among all functions /e E such that /(x) — 1 for a point
x e X where K(x, x) ^ 0, the function K(. , x)/K(x, x) has the smallest norm.
(d) If X is a topological space and if all the functions /e E are continuous in X, then
the functions K( . , x) (where x takes all values in X, or in a dense subset of X) form a
total subset of E (show that there is no element h ^ 0 in E which is orthogonal to all
the elements K(., x)). In particular, if X is a separable metric space, E is a separable
Hilbert space.

5. (a) The notations being those of Problem 4, in order that there exist a reproducing
kernel for E, a necessary and sufficient condition is that for every x e X, the linear form
/-»•/(#) be continuous in E. The reproducing kernel is then unique.

(b) Deduce from (a) that if there exists a reproducing kernel for E, there also exists a
reproducing kernel for every closed vector subspace EJ. of E. If Ki is the reproducing
kernel for EI, show that for every function /e E, the function y-*(f\ Ki(. , j>)) is the
orthogonal projection of /in EI. If E2 is the orthogonal supplement of EI and K2 the
reproducing kernel for E2 , then Ki + K2 is the reproducing kernel for E.

6. Let X be a set, E a vector subspace of Cx, on which is given a structure of complex
prehilbert space. In order that there should exist a Hilbert space £ <=• Cx containing E,
such that the scalar product on E is the restriction of the scalar product on 6, and that
there exists a reproducing kernel for £, it is necessary and sufficient that E satisfy the
two following conditions: (1) for every x eX, the linear form f-+f(x) is continuous
in E; (2) for any Cauchy sequence (/„) in E such that, for every x e X, lim fn(x) == 0,

ft-* 00

one has lim ||/n|| = 0. (To prove the conditions are sufficient, consider the subspace £

H-+OQ

of Cx whose elements are the functions /for which there exists a Cauchy sequence (fn)
in E such that lim /„(;*:) =/(#) for every x e X. Show that, for all Cauchy sequences (/„)

n~».oo

having that property, the number lim ||/rt|| is the same, and if ||/|| is that number, this

n~»oo

defines on £ a structure of normed space which is deduced from a structure of prehilbert